Encounter theory

The relative success of the binary encounter and Bethe theories, and the relatively well established systematic trends observed in the measured differential cross sections for ionization by fast protons, has stimulated the development of models that can extend the range of data for use in various applications. It is clear that the low-energy portion of the secondary electron spectra are related to the optical oscillator strength and that the ejection of fast electrons can be predicted reasonable well by the binary encounter theory. The question is how to merge these two concepts to predict the full spectrum. 

A simple alternative derivation of Eq. 9-30 was developed by Dexter French,34 who also provided the author with most of this discussion of encounter theory. Consider a small element of volume AV swept out by a particle as it moves through the solution for a distance equal to its own radius. This element of volume will equal nr3 ... 

However, the very first attempt to justify DET starting from the general multiparticle approach to the problem led to a surprising result it revealed the integral kinetic equation rather than differential one [33], This equation constitutes the basis of the so-called integral encounter theory (IET), which is a kind of memory function formalism often applied to transport phenomena [34] or spectroscopy [35], but never before to chemical kinetics. The memory... 

The remote transfer in condensed matter is characterized by the position-dependent rate W(r), which is the input data for encounter theory. In its differential version (DET), the main kinetic equation (3.2) remains unchanged, but the rate constant acquires the definition relating it to W(r) ... 

Figure 3.8. (a) The linear viscosity dependence of the inverse ionization rate in the reaction studied in Ref. 98. Bullets—experimental points solid line—fit performed with the generalized Collins—Kimball model, (b) The effective quenching radius for the same reaction in the larger range of the viscosity variation. Bullets—experimental points solid fine—fit performed with the encounter theory for the exponential transfer rate. The diffusion coefficient D given in A2/ns was calculated from the Stokes—Einstein relationship corrected by Spemol and Wirtz [100]. 

This is a clear demonstration that a differential encounter theory is not a reasonable alternative to an integral one, when the energy transfer is reversible and occurs between the unstable reactants. Since the lifetimes are not equal, the rate constant of the energy transfer from the short-lived to the long-lived particle does not exist (diverges) at long times. In contrast, in integral formalism we do not encounter any difficulties. This is illustrated by the simplest example of contact transfer considered below. 

A different situation arises if one calculates the same constant in the superposition approximation (SA) competing with encounter theory. This approximation was first employed in Refs. 139 and 140 and then applied to a number of particular problems [141-145], However, the present problem is an exceptional one because it has an exact solution that allows the examination of all others. To make this possible, the original SA was employed for this very problem in Ref. 136, and the following Stem-Volmer constant was obtained ... 

In this respect the SA equations are rather unnatural and their distribution functions pA and p/ describe pairs surrounded by other particles rather than isolated pairs as n and m in encounter theory. On the other hand, the macroscopic SA equations include the pumping term 7(f) that allows for then-stationary solution, which is incorrect in differential encounter theory. The accuracy of this solution inspected in Ref. 133 was found to be satisfactory for the irreversible quenching. Unfortunately, for the reversible transfer, this is not the case. 

Although small, this is a principal disadvantage of the simplest integral theory. The near-contact density of the products nonlinear in c is lost in the lowest-order approximation to this parameter. However, the nonzero contribution to this region is provided by a modified encounter theory outlined in Section XII. The chief merit of MET is that the argument of the Laplace transformation of n r,t) in (3.311) is shifted from 1 /td to 1/xd + ck. As a result, in the limit xD = oo we have instead of (3.313) [133] ... 

We have seen in Section IV that the study of the reversible reaction of energy transfer was made possible only by means of integral encounter theory. The same is true for reversible electron transfer (3.354) that was first considered with IET in Ref. 188 and then in a much wider context in subsequent publications [107,189],... 

This behavior, inherent to the IET description of either reversible or irreversible transfer, can be eliminated using modified integral encounter theory (MET) [41,44], or an improved superposition approximation [51,126],... 

At the same time we do not have to assume that A -C c as we did previously. Under this condition the acceptor concentration A = [A] remained almost constant, approximately equal to its initial value c. In what follows we will eliminate this restriction and account for the expendable neutral acceptors whose concentration A(t) decreases in the course of ionization. When there is a shortage of acceptors, the theory becomes nonlinear in the concentration, even in absence of bulk recombination. Under such conditions only general encounter theories are appropriate for a full timescale (non-Markovian) description of the system relaxation. We will compare them against each other and with the properly generalized Markovian and model theories of the same phenomena. 

Vectors (3.493) obey the following general equations of integral encounter theory ... 

The modified encounter theory is the only one that gives dilferent forms for S(j) in the trap (Table VIII) and target (Table IX) limits. In the case of the... 

In the context of a single review, it is difficult to cover all the great diversity of encounter theory applications. Some of the applications that were not covered are listed here ... 

All these applications in conjunction with those included in the present review testify to the fact that encounter theory has been converted to a solo branch of non-Markovian chemical kinetics. 

However, encounter theory is subdivided into IET, MET, and DET/UT, and neither of them is perfect. IET is in fact the most universal and general approach to binary reactions in three-dimensional space at low concentration of reactants. Moreover, this is a common limit for all other theories valid at higher concentrations. IET is applied to contact and distant reactions, both reversible and irreversible, and is especially suited for analytic calculations of quantum... 

However, at still larger concentrations only DET/UT is capable of reaching the kinetic limit of the Stem-Volmer constant and the static limit of the reaction product distribution. On the other hand, this theory is intended for only irreversible reactions and does not have the matrix form adapted for consideration of multistage reactions. The latter is also valid for competing theories based on the superposition approximation or nonequilibrium statistical mechanics. Moreover, most of them address only the contact reactions (either reversible or irreversible). These limitations strongly restrict their application to real transfer reactions, carried out by distant rates, depending on the reactant and solvent parameters. On the other hand, these theories can be applied to reactions in one- and two-dimensional spaces where binary approximation is impossible and encounter theories inapplicable. 

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